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On the de Almeida--Thouless Transition Surface in the Multi-Species SK Model with Centered Gaussian External Field

Heejune Kim

Abstract

We study the phase transition of the Parisi formula for the free energy in the multi-species Sherrington--Kirkpatrick model with a centered Gaussian external field and a positive-semidefinite variance profile matrix. We show that in terms of the strength of the external field and the variance profile, the de Almeida--Thouless surface delineates the boundary between replica symmetric solutions and replica symmetry breaking solutions.

On the de Almeida--Thouless Transition Surface in the Multi-Species SK Model with Centered Gaussian External Field

Abstract

We study the phase transition of the Parisi formula for the free energy in the multi-species Sherrington--Kirkpatrick model with a centered Gaussian external field and a positive-semidefinite variance profile matrix. We show that in terms of the strength of the external field and the variance profile, the de Almeida--Thouless surface delineates the boundary between replica symmetric solutions and replica symmetry breaking solutions.

Paper Structure

This paper contains 22 sections, 49 theorems, 297 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Overview of the proof
  4. Organization of the paper
  5. Minimizers of the replica symmetric functional
  6. Proof of Proposition \ref{['prop: uniqueness of fixed-point']}
  7. Proof of Proposition \ref{['prop: infimum of RS functional for positive-definite']}
  8. Further properties of fixed-points
  9. Support of Parisi measures
  10. Preparatory lemmas
  11. Proof of Proposition \ref{['prop: min and max of support']}
  12. Interpolation and the Aizenman--Sims--Starr scheme
  13. The ASS scheme along the interpolation
  14. Proof of Proposition \ref{['prop: sufficient condition for RS']}
  15. Establishing a free energy cost for the coupled free energy
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any irreducible positive-semidefinite $\Delta^2$ and for any $\tau^2 \succeq\mymathbb{0}$, the replica symmetric solution holds, that is, if and only if $(\Delta^2,\tau^2)$ lies inside the AT surface, $\rho(\Gamma\Delta^2\Lambda)\le 1/2.$

Figures (1)

  • Figure 1: Illustration of Step 4 in two-species. Three blue dots indicate $\operatorname{supp}(\mu)$ which is totally ordered with respect to the partial order $\preceq$. The smaller ellipse depicts $\mathsf{B}(q^*,\sqrt{\delta(t)})$ and the larger ellipse depicts $\mathsf{B}(q^*, C' \sqrt{\delta(t)})$. The first and the third quadrants shaded in light blue depicts $\mathsf{T}= \{q\in [0,1]^{\mathscr S}: q\preceq q^* \; \text{or} \; q\succeq q^*\}$. Although $\{q^\mu_{\min}, q^\mu_{\max}\} \subseteq \{q^*\}\cup \bigl(\operatorname{int}(\mathsf{T})\cap \operatorname{cl}(\mathsf{B}(q^*,\sqrt{\delta(t)}))\bigr)$, there remains a blue dot that lies outside $\operatorname{cl}(\mathsf{B}(q^*,\sqrt{\delta(t)}))$, which necessitates enlarging the ellipse to ensure containment. The ellipses are always elongated in this way, since the principal eigenvector (red dashed arrow) of $\Lambda\Delta^2\Lambda$ has positive entries by the Perron--Frobenius theorem.

Theorems & Definitions (97)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Proposition \ref{['prop: uniqueness of fixed-point']}
  • Lemma 2.6
  • Remark 2.7
  • ...and 87 more